Question

I need to explain, using the limit formula (f(a+h)-f(a))/h and a picture (graph), as to why it gives the slope of the tangent line to f(x) at a. Can anyone help?

Answer 1

Imagine some curve \(f(x)\) that's continuous for some domain (it doesn't matter which), and take two points on that curve, \((x_1,y_1)\) and \((x_2,y_2)\), with \(x_1\) and \(x_2\) within that interval. The slope of the line segment connecting the two points is
\[\frac{y_2-y_1}{x_2-x_1}\]
In terms of the function,
\[\text{slope}=\frac{f(x_2)-f(x_1)}{x_2-x_1}\]
Obviously, the two points must be distinct (\(x_1\not=x_2\)); otherwise, the slope would be undefined.

Since they are distinct, you can think of \(x_2\) as \(x_1\) plus some difference: \(x_2=x_1+h\). Rearranging it a bit, you have \(h=x_2-x_1\).

Note that as \(h\to0\), the difference between these two points gets smaller and smaller.

Substituting these into the slope formula, you get
\[\frac{f(x_1+h)-f(x_1)}{h}\]
Let's call this remaining value of x \(a\):
\[\frac{f(a+h)-f(a)}{h}\]
Taking the limit thus gives you a closer and closer approximation of what the tangent's slope is. Thus, you have
\[\text{slope}=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\]